Given

• Orders: $𝑝=1$, $𝑑=1$, $𝑞=1$, $𝐾=2$

• Coefficient matrices:

  $${𝚽}_1=\left[\begin{array}{cc}0.4& 0.2\\ 0.2& 0.4\end{array}\right]{𝚯}_1=\left[\begin{array}{cc}0.4& 0.2\\ 0.2& 0.4\end{array}\right]$$
• Intercept: $𝒄=\left[\begin{array}{c}0\\ 0\end{array}\right]$
• Initial conditions: ${𝒅}_1=\left[\begin{array}{c}0\\ 0\end{array}\right]$, ${𝜺}_1=\left[\begin{array}{c}0\\ 0\end{array}\right]$
• Two series stacked into ${𝒙}_{𝑡}=\left[\begin{array}{c}{𝑥}_{1,𝑡}\\ {𝑥}_{2,𝑡}\end{array}\right]$:

Step 1 — formula

Substitute orders into the VARIMA recursion:

Difference — apply $\left(1-𝐵\right)$ component-wise:

In differenced space, the model is VARMA$\left(1,1\right)$:

Forecast the difference (set ${𝜺}_{𝑡}=\mathrm{𝟎}$):

Undifference — convert back to a forecast for ${𝒙}_{𝑡}$:

Innovation:

Step 2 — apply at $𝑡=2$

First difference: ${𝒅}_2={𝒙}_2-{𝒙}_1=\left[\begin{array}{c}10\\ 9\end{array}\right]-\left[\begin{array}{c}12\\ 8\end{array}\right]=\left[\begin{array}{c}-2\\ 1\end{array}\right]$.

Plug in ${𝚽}_1$, ${𝚯}_1$, ${𝒅}_1=\left[\begin{array}{c}0\\ 0\end{array}\right]$, ${𝜺}_1=\left[\begin{array}{c}0\\ 0\end{array}\right]$:

Step 3 — iterate

Pipeline at each $𝑡$: difference $\to$ VARMA-forecast $\to$ undifference. Values rounded to 4 decimal places.